Thanks for your explanation, but my question is more about ""jemanden/jemandem telefonieren" existance,than about "anrufen" and "telefonieren" differences.
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user508Jun 23 '11 at 7:29

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Gigli, you asked "Is it correct to use 'telefonieren' like 'anrufen'?". That is the only question in your narrative. You did NOT ask whether 'jemanden/jemandem telefonieren" is correct, at least, I do not see that in the question. In any case, it is "mit jemandem telefonieren" (Dativobjekt) and "jemanden anrufen"(Akkusativobjekt). At least in Standard German. Apparently the Swiss have variants, but these sound VERY awkward to a non-Swiss person.
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teylynJun 23 '11 at 8:54

@teylyn: You have a point here. That's due to me being Swiss ;) Have a look here: blogwiese.ch/archives/584 It's strange though, when I think about the infinitive, I would always think of "jemaden anrufen", i.e. Akkusativ as in "dich anrufen" - my mind works in strange ways, I guess.
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phant0mJun 23 '11 at 1:09

" Also, "Wir haben gestern über eine Stunde [lang] telefoniert." This sentence can change meaning depending on how the words are stressed. "Wir haben gestern über eine STUNDE telefoniert." That will be talking about a lesson. "Wir haben gestern ÜBER eine Stunde telefoniert." That will be a phone call that lasted more than an hour." Yes, that is already written in my answer.
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phant0mJun 23 '11 at 1:16

As others have noted, there is also the slight difference in meaning that you have already done the action of "anrufen" if the other person did not answer their phone, but for "telefonieren" there should actually be communication.

Telefonieren is specifically to (talk over a) "telephone." "Anrufen" is to call someone. The normal implication is via a "telephone." But in theory, it could be through radio, internet, walkie-talkie, or even across a room.

If I say "Ich ruf dich an", it's certainly not across a crowded room. There are several meanings to "anrufen", e.g. call a court of law, but in everyday language these would not be very commen.
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teylynJun 23 '11 at 1:07